Question

Consider a population of 1024 mutual funds that primarily invest in large companies. You have determined that population mean, the mean? one-year total percentage return achieved by all the? funds, is 8.80 and that sigma?, the standard? deviation, is 0.50. Complete? (a) through? (c).

a. According to the empirical? rule, what percentage of these funds is expected to be within ?±2 standard deviations of the? mean? ?

b. According to the Chebyshev? rule, what percentage of these funds are expected to be within ?±8 standard deviations of the? mean? ?(Round to two decimal places as? needed.)

c. According to the Chebyshev? rule, at least 93.75?% of these funds are expected to have? one-year total returns between what two? amounts? Between ____ and ____. ?(Round to two decimal places as? needed.

Answer 1

a) According to the empirical rule, 68% lies within 1 standard deviation from the mean, 95% within 2 standard deviations from mean and 99.7% within 3 standard deviations from mean. Therefore 95% is the correct answer here.

b) According to Chebyshev's inequality theorem, at least 1 - 1/k² of the observations lies within k standard deviations from the mean. Therefore 1 - 1/8² = 63/64 = 98.44%

Therefore at least 98.44% of the observations lies within 8 standard deviations from the mean.

c) Here, we simply need to find k

1 - 1/k² = 0.9375

1/k² = 0.0625

k = 4

Therefore the limits here are computed as:
Mean - 4*Std Dev = 8.8 - 4*0.5 = 6.80 and Mean + 4*Std Dev = 10.80

Therefore between 6.80% and 10.80%